A Note on <Emphasis Type="Italic">b</Emphasis>-Weakly Compact Operators

We consider a continuous operator T: E → X where E is a Banach lattice and X is a Banach space. We characterize the b-weak compactness of T in terms of its mapping properties.     

Weakly compact operators on non-complete normed spaces

We prove that weakly compact operators on a non-reflexive normed space cannot be bijective. We also show that, in the above result, bijectivity cannot be relaxed to surjectivity. Finally, we study the behaviour of surjective weakly compact operators on a non-reflexive normed space, when they are perturbed by small scalar multiples of the identity, and derive from this study the recent result of Spurný [A note on compact operators on normed linear spaces, Expo. Math. 25 (2007) 261–263] that compact operators on an infinite-dimensional normed space cannot be surjective.     

Spectral Exponent of Finite Sums of Weighted Positive Operators in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-Spaces

In this paper we investigate the spectral exponent, i.e. logarithm of the spectral radius of operators having the form

Optimal Domains and Integral Representations of Convolution Operators in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(<Emphasis Type="Italic">G</Emphasis>)

Given , a compact abelian group G and a function , we identify the maximal (i.e. optimal) domain of the convolution operator (as an operator from L^p(G) to itself). This is the largest Banach function space (with order continuous norm) into which L^p(G) is embedded and to which has a continuous extension, still with values in L^p(G). Of course, the optimal domain depends on p and g. Whereas is compact, this is not always so for the extension of to its optimal domain. Several characterizations of precisely when this is the case are presented.     

ISOLATED POINTS OF THE APPROXIMATE POINT SPECTRUM OF CERTAIN LATTICE HOMOMORPHISMS ON Co (X)

Abstract

Suppose X is a locally compact Hausdorff space and C (X) the apace of all continuous complex valued functions on X which vanish at infinity. Let T be a (complex) linear lattice homomorphism on C_o (X) whose adjoint is also a lattice homomorphism. It is sham that every non-zero isolated point of the approximate point spectrum of T lies in the point spectrum of T. An example is given to show that the exclusion of zero is necessary, even when X is compact. The same techniques are then used to show that if also the spectrum of T is finite then T can be written, in a natural manner, as a direct sum of two such lattice homomorphisms; one being an n'th root of an invertible multiplication operator and the other quasi-nilpotent.     

Hypercyclic Conjugate Operators

We prove that for any weighted backward shift B = B_w on an infinite dimensional separable Hilbert space H whose weight sequence w = (w_n) satisfies , the conjugate operator is hypercyclic on the space S(H) of self-adjoint operators on H provided with the topology of uniform convergence on compact sets. That is, there exists an such that is dense in S(H). We generalize the result to more general conjugate maps , and establish similar results for other operator classes in the algebra B(H) of bounded operators, such as the ideals K(H) and N(H) of compact and nuclear operators respectively.     

Normality of Products of <Emphasis Type="Italic">p</Emphasis>-Hyponormal Operators and Their Aluthge Transforms

Let B(H) denote the algebra of operators on a complex separable Hilbert space H, and let A $\in$ B(H) have the polar decomposition A = U|A|. The Aluthge transform is defined to be the operator . We say that A $\in$ B(H) is p-hyponormal, . Let . Given p-hyponormal , such that AB is compact, this note considers the relationship between denotes an enumeration in decreasing order repeated according to multiplicity of the eigenvalues of the compact operator T (respectively, singular values of the compact operator T). It is proved that is bounded above by and below by for all j = 1, 2, . . . and that if also is normal, then there exists a unitary U¹ such that for all j = 1, 2, . . ..     

On principal T-bands in a Banach lattice

It is shown that for every positive order continuous Riesz operatorT, defined on an order complete complex Banach latticeE which is separated by its Köthe dual, there exists a Frobenius decomposition ofE into a countable number of disjoint principalT-bands and a band on whichT is quasi-nilpotent. A basis for the generalized eigenspace ofT pertaining to its maximal eigenvalue is constructed and the positivity properties of its elements are studied. The distinguished eigenvalues ofT are characterized and it is also shown that the theory ofT-bands is symmetric with respect to the duality which exists betweenE and its Köthe dual. This generalizes aspects of work done by H.D. Victory and R.-J. Jang-Lewis.     

Compact and weakly compact weighted composition operators on weighted spaces of continuous functions

In this note we characterize the compact and weakly compact weighted composition operatorsW _, on certain weighted locally convex spacesCV _o(X, E) of vector-valued continuous functions induced by self maps ofX and the operator-valued mappings XB(E).The work of this author was supported in part by CSIR Grant 9/100/92-EMR-IThe work of this author was supported in part by UGC Grant F.8-7/91 (RBB-II)     

On inequalities for eigenvalues of 2 × 2 matrices with Schatten–von Neumann entries

Let SN_r (r ≥ 1) denote the Schatten-von Neumann ideal of compact operators in a separable Hilbert space. For the block matrix

the inequality

(p = 2; 3;?…?) is proved, where λ_k(A) (k = 1; 2;?…?) are the eigenvalues of A and N_r(.) is the norm in SN_r. Moreover, let P(z) = z²I + Bz + C (z ∈ ?) with B ∈ SN_2p, C ∈ SN_p. By z_k(P) (k = 1; 2;?…?) the characteristic values of the pencil P are denoted. It is shown that

In the case p = 1, sharper results are established. In addition, it is derived that

     

Inequalities for sums and direct sums of Hilbert space operators

We prove several singular value inequalities and norm inequalities involving sums and direct sums of Hilbert space operators. It is shown, among other inequalities, that if X and Y are compact operators, then the singular values of are dominated by those of X ⊕ Y. Applications of these inequalities are also given.     

Regularized spectral distributions

For C a bounded, injective operator with dense image, we define a C-regularized spectral distribution. This produces a functional calculus, f f(B), from C() into the space of closed densely defined operators, such that f(B)C is bounded when f has compact support. As an analogue of Stone's theorem, we characterize certain regularized spectral distributions as corresponding to generators of polynomially bounded C-regularized groups. We represent the regularized spectral distribution in terms of the regularized group and in terms of the C-resolvent. Applications include the Schrödinger equation with potential, and symmetric hyperbolic systems, all on L^p(ⁿ) (1p<), C _o(ⁿ), BUC(ⁿ), or any space of functions where translation is a bounded strongly continuous group.     

Multilinear Forms of Hilbert Type and Some Other Distinguished Forms

We give some new examples of bounded multilinear forms on the Hilbert spaces ℓ₂ and L₂ (0, ∞). We characterize those which are compact or Hilbert-Schmidt. In particular, we study m-linear forms (m ≥ 3) on ℓ₂ which can be regarded as the multilinear analogue of the famous Hilbert matrix. We also determine the norm of the permanent on where      

Orthogonality in Classes

 Let denote the von Neumann–Schatten class, its norm and let be an elementary operator defined by . We shall characterize those operators which are orthogonal to the range of in the sense that for all . The main results of this paper are: If and (i) if A, C, respectively B, D are commuting normal operators with , or (ii) if A, B are contractions and , then is orthogonal to the range of if and only of S is in the kernel of . Furthermore, in both cases, the algebraic direct sum satisfies . (Received 9 February 2000; in revised form 21 February, 2001)     

Some Properties of Essential Spectra of a Positive Operator

Let E be a Banach lattice, T be a bounded operator on E. The Weyl essential spectrum σ_ew(T) of the operator T is a set , where is a set of all compact operators on E. In particular for a positive operator T next subsets of the spectrum

SOME FUNCTIONAL ANALYTIC PROPERTIES OF COMPOSITION OPERATORS

Abstract

This is a report on a number of recent results on composition operators which map, for 0 < p ? q ∞, the Hardy space H^p (on the unit disk in the complex plane) into H ^q. Attention is focused on questions of boundedness (existence), compactness, order boundedness and, in connection with the latter, on relating the absolutely summing and nuclearity character as well as special factorization properties of the operator to function theoretic properties of the defining symbol. Moreover, tools are provided to show that certain classes of operators can well be distinguished already on the level of composition operators.     

Norm Inequalities for Sums of Positive Operators. II

It is shown that if A and B are positive operators on a separable complex Hilbert space, then for every unitarily invariant norm. When specialized to the usual operator norm ||·|| and the Schatten p-norms ||·||_p, this inequality asserts that and These inequalities improve upon some earlier related inequalities. Other norm inequalities for sums of positive operators are also obtained.     

Some results about operators on L1

Let (Ω, Σ, µ) be a probability space. We give a few results about operators on L₁(µ). Among these, we show that if T is a bounded linear operator on L₁(µ) which acts as a Hilbert-Schmidt operator on L₂(µ), then T : L₁(µ) → L₁(µ) is representable.     

Uniformly summing sets of operators on spaces of vector–valued continuous functions

Let X, Y be Banach spaces. We say that a set is uniformly p–summing if the series is uniformly convergent for whenever (x_n) belongs to . We consider uniformly summing sets of operators defined on a -space and prove, in case X does not contain a copy of c₀, that is uniformly summing iff is, where T (φ x) = (T^#φ) x for all and x∈X. We also characterize the sets with the property that is uniformly summing viewed in . Received: 1 July 2005